High frequency repeated games with costly monitoring

We study two-player discounted repeated games in which one player cannot monitor the other unless he pays a fixed amount. It is well known that in such a model the folk theorem holds when the monitoring cost is on the order of magnitude of the stage payoff. We analyze high frequency games in which the monitoring cost is small but still significantly higher than the stage payoff. We characterize the limit set of public perfect equilibrium payoffs as the monitoring cost tends to 0. It turns out that this set is typically a strict subset of the set of feasible and individually rational payoffs. In particular, there might be efficient and individually rational payoffs that cannot be sustained in equilibrium. We also make an interesting connection between games with costly monitoring and games played between long-lived and short-lived players. Finally, we show that the limit set of public perfect equilibrium payoffs coincides with the limit set of Nash equilibrium payoffs. This implies that our characterization applies also to sequential equilibria.